D

Full Chapter Overview

1.1 Variables

What are Variables?

A variable is a placeholder for an unknown or general value. It can represent a single number, multiple numbers, or a general category of elements. Think of variables as mathematical placeholders—they temporarily stand in for values we do not yet know.

Why Use Variables?

Unknown Values:

When solving equations, a variable represents an unknown. Example: "Is there a number that, when doubled and added to 3, equals its square?" This can be written as: 2x + 3 = x²

General Statements:

Variables help express general rules. Example: Instead of saying, "The square of any even number is even," we generalize: For any even integer n, n² is even.

Types of Mathematical Statements

Mathematical statements often fall into these categories:

  • Universal Statements: Apply to all elements in a set. Example: "All positive numbers are greater than zero."

  • Conditional Statements: Express if-then logic. Example: "If a number is divisible by 18, then it is divisible by 6."

  • Existential Statements: Assert that at least one example exists. Example: "There exists a prime number that is even."

  • Universal Conditional Statements: A universal conditional statement is a statement that is both universal and conditional. Example: For every animal a, if a is a dog, then a is a mammal.

  • Universal Existential Statements: A universal existential statement is a statement that is universal because its first part says that a certain property is true for all objects of a given type, and it is existential because its second part asserts the existence of something. Example: Every real number has an additive inverse.

  • Existential Universal Statements: An existential universal statement is a statement that is existential because its first part asserts that a certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind. Example: There is a positive integer that is less than or equal to every positive integer.

How to Use Variables

  1. Choose a letter (e.g., x, y, z, n, m) to represent the variable.

  2. Clearly define what the variable represents (e.g., "Let x be a real number").

  3. Substitute the variable into equations or expressions.Example: Solve for x:

    • 3x + 5 = 14

    • Subtract 5 from both sides: 3x = 9

    • Divide by 3: x = 3

Summary

Variables are essential for expressing mathematical ideas precisely. They help us solve problems and make general statements efficiently.

1.2 The Language of Sets

What is a Set?

A set is a collection of distinct objects, called elements or members. The order of elements in a set does not matter.

Representing Sets

  • Set-Roster Notation: Listing elements inside curly braces.Example: A = {1, 2, 3}

  • Set-Builder Notation: Describes properties elements must satisfy.Example: B = {x | x is a positive real number}

Special Sets of Numbers

  • R: The set of all real numbers (rational + irrational numbers).

  • Z: The set of all integers {..., -2, -1, 0, 1, 2,...}.

  • Q: The set of all rational numbers (fractions).

  • N: The set of all positive integers {1, 2, 3,...}.

  • Empty Set: The empty set is a set that has no elements. It is denoted by the symbol ∅ or {}.

Relationships Between Sets

  • Subset (⊆): A ⊆ B means every element in A is in B.

  • Proper Subset (⊂): A ⊂ B means A is a subset of B but not equal to B.

  • Equality (=): Two sets are equal if they contain the exact same elements.

Operations on Sets

  • Union (∪): A ∪ B contains all elements in A or B (or both).

  • Intersection (∩): A ∩ B contains elements that are in both A and B.

  • Difference (A - B): Elements in A but not in B.

  • Cartesian Product (A × B): The set of all ordered pairs (a, b), where a is in A and b is in B.Example: If A = {1, 2} and B = {x, y}, then: A × B = {(1, x), (1, y), (2, x), (2, y)}

Distinction between ∈ and ⊆

  • ∈: The symbol ∈ means "is an element of". It is used to show that an object is a member of a set. Example: 1 ∈ {1, 2, 3}

  • ⊆: The symbol ⊆ means "is a subset of". It is used to show that one set is contained in another set. Example: {1, 2} ⊆ {1, 2, 3}

Summary

Sets are foundational in mathematics. Understanding set notation, special sets, and set operations is crucial for logical reasoning.

1.3 The Language of Relations and Functions

What is a Relation?

A relation between two sets A and B is a set of ordered pairs (a, b), where a is in A and b is in B. This defines a relationship between elements of the sets.

Functions

A function is a special relation where each element in A is related to exactly one element in B.

Function Machines

Functions can be thought of as machines that take an input, process it, and produce a single output.

Arrow Diagram of a Relation

An arrow diagram is a visual way to represent a relation between two sets. The elements of the sets are represented as points, and arrows are drawn between the points to indicate the relation.

Types of Functions

  • Injective Function: A function is injective if each element in the co-domain is mapped to by at most one element in the domain. (ie: R = R)

    • each output has a maximum of one input

  • Surjective Function: A function is surjective if each element in the co-domain is mapped to by at least one element in the domain. (ie: R = R²)

    • each output has at least one input but can have more than one

  • Bijective Function: A function is bijective if it is both injective and surjective. (ie: R = R³)

    • This means that for every element in the co-domain, there exists a unique element in the domain that maps to it, ensuring a perfect one-to-one correspondence between the two sets.

Summary

Relations define general connections between sets, while functions ensure each input maps to exactly one output.

1.4 The Language of Graphs

What is a Graph?

A graph consists of:

  • Vertices (Nodes): Points representing elements.

  • Edges: Lines connecting vertices.

Types of Graphs

  • Directed Graph: Edges have a direction (arrows).

  • Undirected Graph: Edges have no direction.

  • Weighted Graph: Edges have a weight associated with them.

Graphs as Relations

Graphs can visually represent relations by using:

  • Vertices for elements.

  • Edges to indicate relations.

Ways to Represent a Graph

  • Adjacency Matrix: A matrix that stores the weights of the edges between vertices.

  • Adjacency List: A list that stores the neighbors of each vertex.

Summary

Graphs provide a visual way to represent relationships, making patterns easier to analyze.